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A110475 Number of symbols '*' and '^' to write the canonical prime factorization of n.

%I A110475 #24 Mar 10 2025 11:01:33
%S A110475 0,0,0,1,0,1,0,1,1,1,0,2,0,1,1,1,0,2,0,2,1,1,0,2,1,1,1,2,0,2,0,1,1,1,
%T A110475 1,3,0,1,1,2,0,2,0,2,2,1,0,2,1,2,1,2,0,2,1,2,1,1,0,3,0,1,2,1,1,2,0,2,
%U A110475 1,2,0,3,0,1,2,2,1,2,0,2,1,1,0,3,1,1,1,2,0,3,1,2,1,1,1,2,0,2,2,3,0,2,0,2,2
%N A110475 Number of symbols '*' and '^' to write the canonical prime factorization of n.
%C A110475 It is conjectured that 1,2,3,4,5,6,7,9,11 are the only positive integers which cannot be represented as the sum of two elements of indices n such that a(n) = 1. - _Jonathan Vos Post_, Sep 11 2005
%C A110475 a(n) = 2 iff n is a sphenic number (A007304) or n is a prime p times a prime power q^e with e > 1 and q not equal to p. a(n) = 3 iff n has exactly four distinct prime factors (A046386); or n is the product of two prime powers (p^e)*(q^f) with e > 1, f > 1 and p not equal to q; or n is a semiprime s times a prime power r^g with g > 1 and r relatively prime to s. For a(n) > 3, _Reinhard Zumkeller_'s description is a simpler description than the above compound descriptions. - _Jonathan Vos Post_, Sep 11 2005
%H A110475 Reinhard Zumkeller, <a href="/A110475/b110475.txt">Table of n, a(n) for n = 1..10000</a>
%H A110475 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeFactorization.html">Prime Factorization</a>.
%F A110475 a(n) = A001221(n) - 1 + A056170(n) for n > 1.
%F A110475 a(n) = 0 iff n=1 or n is prime: a(A008578(n)) = 0.
%F A110475 a(n) = 1 iff n is a semiprime or a prime power p^e with e > 1.
%F A110475 From _Amiram Eldar_, Sep 27 2024: (Start)
%F A110475 a(n) = A238949(n) - 1 for n >= 2.
%F A110475 Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B + C - 1), where B is Mertens's constant (A077761) and C = Sum_{p prime} 1/p^2 (A085548). (End)
%e A110475 a(208029250) = a(2*5^3*11^2*13*23^2) = 4 '*' + 3 '^' = 7.
%t A110475 A110475[n_] := 2*Length[#] - 1 - Count[#, 1] & [FactorInteger[n][[All, 2]]];
%t A110475 Array[A110475, 100] (* _Paolo Xausa_, Mar 10 2025 *)
%o A110475 (Haskell)
%o A110475 a110475 1 = 0
%o A110475 a110475 n = length us - 1 + 2 * length vs where
%o A110475             (us, vs) = span (== 1) $ a118914_row n
%o A110475 -- _Reinhard Zumkeller_, Mar 23 2014
%Y A110475 Cf. A050252, A001358, A025475, A000040, A238949.
%Y A110475 Cf. A007304, A046386, A008578.
%Y A110475 Cf. A001221, A056170, A118914, A077761, A085548.
%K A110475 nonn
%O A110475 1,12
%A A110475 _Reinhard Zumkeller_, Sep 08 2005